QED and relativistic corrections in superheavy elements

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Paul Indelicato, J.P. Santos, S. Boucard, J.P. Desclaux

To cite this version:

Paul Indelicato, J.P. Santos, S. Boucard, J.P. Desclaux. QED and relativistic corrections in superheavy

elements. The European Physical Journal D : Atomic, molecular, optical and plasma physics, EDP

Sciences, 2007, 45 (1), pp.155-170. �10.1140/epjd/e2007-00229-y�. �hal-00124391v2�

hal-00124391, version 2 - 31 May 2007

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QED and relativistic corrections in superheavy elements

P. Indelicato ¹ , J.P. Santos ² , S. Boucard ¹ , and J.-P. Desclaux ³

1

Laboratoire Kastler Brossel, ´ Ecole Normale Sup´erieure; CNRS; Universit´e P. et M. Curie - Paris 6 Case 74; 4, place Jussieu, 75252 Paris CEDEX 05, France e-mail: paul.indelicato@spectro.jussieu.fr

2

Centro de F´ısica At´ omica and Departamento de F´ısica, Faculdade de Ciˆencias e Tecnologia, Universidade Nova de Lisboa, Monte de Caparica, 2829-516 Caparica, Portugal

3

15 Chemin du Billery, 38360 Sassenage, France e-mail: jean-paul.desclaux@wanadoo.fr

Received: May 31, 2007/ Revised version: date

Abstract. In this paper we review the different relativistic and QED contributions to energies, ionic radii, transition probabilities and Land´e g-factors in super-heavy elements, with the help of the MultiConfigura- tion Dirac-Fock method (MCDF). The effects of taking into account the Breit interaction to all orders by including it in the self-consistent field process are demonstrated. State of the art radiative corrections are included in the calculation and discussed. We also study the non-relativistic limit of MCDF calculation and find that the non-relativistic offset can be unexpectedly large.

PACS. 31.30.Jv – 31.25.Eb – 31.25.Jf – 32.70.Cs

1 Introduction

In the last decades, accelerator-based experiments at GSI and Dubna have lead to the discovery of super-heavy ele- ments up to Z = 116 and 118 [1] (for a recent review see [2]). Considerable theoretical work has been done to pre- dict the ground configuration and the chemical properties of those superheavy elements. Relativistic Hartree-Fock has been used to predict the ground configuration proper- ties of superheavy elements up to Z = 184 in the early 70’s [3,4]. The Multiconfiguration Dirac-Fock (MCDF) method was used to predict orbital properties of elements up to Z = 120 [5], electron binding energies up to Z = 118 [6,7], and K-shell and L-shell ionization potentials for the superheavy elements with Z = 112, 114, 116, and 118 [8]. Ionization potential and radii of neutral and ionized bohrium (Z = 107) and hassium (Z = 108) have been evaluated with large scale MCDF calculations [9]. Kaldor and coworkers have employed the relativistic coupled-cluster method to predict ground state configuration, ionization potential, electron affinity, binding energy of the negative ion of several elements with Z ≥ 100 [10,11,12,13,14,15, 16,17].

Very recently, laser spectroscopy of several fermium (Z = 100) transitions has been performed, the spectroscopy of nobelium (Z = 102) is on the way [18,19,20], and large scale MCDF calculations of transition energies and rates have been performed by several authors for super- heavy elements with Z = 100 [18,21], Z = 102 [21] and Z = 103 [22], which are in reasonable agreement with the fermium measurements.

Send offprint requests to: P. Indelicato

There are however many unanswered questions, that need to be addressed in order to assess the accuracy and the limit of current theoretical methods. For inner-shells, or highly ionized systems, QED effects must be very strong in superheavy elements, where the atomic number Z ap- proaches the limit Zα → 1 (α = 1/137.036 is the fine structure constant), at which the point-nucleus Dirac equa- tion eigen-energies become singular for all s 1/2 and p 1/2

states (the energy depends on q

j + ¹ ₂ 2

− (Zα) ² , where j is the total angular momentum). This means that all QED calculations must be performed for finite nuclei, and to all orders in Zα. QED calculations for outer-shell are very difficult. At present, only the simplest one-electron, one-loop diagrams can be calculated, using model poten- tials to account for the presence of the other electrons.

The evaluation of many-body effects, that remains large for neutral and quasi-neutral systems, is also made very difficult by the complex structure of these atoms, in which there may be several open shells. In that sense, meth- ods based on Relativistic Many-Body Perturbation theory (RMBPT) and MCDF methods are complementary. The former one usually allowing for more accurate results, but limited to (relativistic) closed shell systems minus one or plus one or at most two electrons, while the latter is com- pletely general but convergence becomes problematic for large size configuration set, particularly if one wants to optimize all orbitals.

The paper is organized as follows. In Sec. 2 we present

briefly the MCDF method, with emphasis on the specific

features of the code we have been using, and we described

the QED corrections that have been used in the calcula-

tions. In Sec. 3 we describes specific problems associated with the MCDF method (or more generally to all-order methods). We thus study the non-relativistic limit of the MCDF codes, and specific problems associated with high- Z . In Sec. 3.2 we study a number of systems from highly- charges ions to neutral atoms, for very large atomic num- bers. The evaluation of atomic charge distribution size and Land´e factors is performed in Sec. 5 and in Sec. 6 we state our conclusion.

2 Calculation of atomic wavefunctions and transition probabilities

2.1 The MCDF method

In this work, bound-states wavefunctions are calculated using the 2006 version of the Dirac-Fock program of J.-P.

Desclaux and P. Indelicato, named mdfgme [23]. Details on the Hamiltonian and the processes used to build the wave-functions can be found elsewhere [24,25,26,27].

The total wavefunction is calculated with the help of the variational principle. The total energy of the atomic system is the eigenvalue of the equation

H

Ψ Π,J,M (. . . , r _i , . . .) = E Π,J,M Ψ Π,J,M (. . . , r _i , . . .), (1) where Π is the parity, J is the total angular momentum eigenvalue, and M is the eigenvalue of its projection on the z axis J z . Here,

H

=

N

X

i=1

H D (r i ) + X

i<j

V(| r _i − r _j |), (2) where H D is the one electron Dirac operator and V is an operator representing the electron-electron interaction of order one in α. The expression of V ij in Coulomb gauge, and in atomic units,is

V ij = 1 r ij

(3a)

− α _i · α _j r ij

(3b)

− α _i · α _j r ij

[cos ω ij r ij

c

− 1]

+ c ² ( α _i · ∇ _i )( α _j · ∇ _j ) cos ^ω

_c ^r

− 1 ω ² _ij r ij

, (3c) where r ij = | r _i − r _j | is the inter-electronic distance, ω ij is the energy of the exchanged photon between the two elec- trons, α _i are the Dirac matrices and c is the speed of light.

We use the Coulomb gauge as it has been demonstrated that it provides energies free from spurious contributions at the ladder approximation level and must be used in many-body atomic structure calculations [28,29].

The term (3a) represents the Coulomb interaction, the term (3b) is the Gaunt (magnetic) interaction, and the last two terms (3c) stand for the retardation operator. In this

expression the ∇ operators act only on r ij and not on the following wavefunctions.

By a series expansion of the operators in expressions (3b) and (3c) in powers of ω ij r ij /c ≪ 1 one obtains the Breit interaction, which includes the leading retardation contri- bution of order 1/c ² . The Breit interaction is, then, the sum of the Gaunt interaction (3b) and the Breit retarda- tion

B ^R _ij = α _i · α _j 2r ij

− ( α _i · r _ij ) ( α _j · r _ij )

2r _ij ³ . (4)

In the many-body part of the calculation the electron- electron interaction is described by the sum of the Coulomb and the Breit interactions. Higher orders in 1/c, deriving from the difference between expressions (3c) and (4) are treated here only as a first order perturbation. All calcu- lations are done for finite nuclei using a Fermi distribution with a tickness parameter of 2.3 fm. The nuclear radii are taken or evaluated using formulas from reference [30].

The MCDF method is defined by the particular choice of a trial function to solve equation (1) as a linear combi- nation of configuration state functions (CSF):

|Ψ ^Π ,J,M i =

n

X

ν=1

c ν |ν, Π , J, M i . (5) The CSF are also eigenfunctions of the parity Π , the total angular momentum J ² and its projection J z . The label ν stands for all other numbers (principal quantum number, ...) necessary to define unambiguously the CSF. The c ν

are called the mixing coefficients and are obtained by di- agonalization of the Hamiltonian matrix coming from the minimization of the energy in equation (1) with respect to the c ν . The CSF are antisymmetric products of one- electron wavefunctions expressed as linear combination of Slater determinants of Dirac 4-spinors

|ν, Π , J, M i =

N

X

i=1

d i

ψ ⁱ ₁ ( r ₁ ) · · · ψ _m ⁱ ( r ₁ ) .. . . .. .. . ψ ⁱ ₁ ( r _m ) · · · ψ ⁱ _m ( r _m )

, (6)

where the ψ-s are the one-electron wavefunctions and the coefficients d i are determined by requiring that the CSF is an eigenstate of J ² and J z . The one-electron wavefunc- tions are defined as

ψ (r) =

χ ^µ _κ (Ω)P (r) iχ ^µ _{− κ} (Ω)Q(r)

, (7)

where χ ^µ _κ is a two-component spinor, and P and Q are respectively the large and small component of the wave- function.

Application of the variational principle leads to a set of

integro-differential equations, which determines the radial

wavefunctions and a Hamiltonian matrix, which provides

the mixing coefficients c ν by diagonalization. The mdfgme

code provide the possibility to obtain wavefunctions and

mixing coefficient with either only the Coulomb (3a) in-

teraction used to obtain the differential equations and the

Hamiltonian matrix that is diagonalized to obtain mixing coefficient or the full Breit operator (4). The convergence process is based on the self-consistent field process (SCF).

For a given set of configurations, initial wavefunctions, ob- tained for example, with a Thomas-Fermi potential, are used to derive the Hamiltonian matrix and set of mixing coefficients. Direct and exchange potential are constructed for all orbitals, and the differential equations are solved.

Then a new set of potentials is constructed and the whole process is repeated. Each time the largest variation of all wavefunction has been reduced by an order of magnitude, a new Hamiltonian matrix is build and diagonalized, and a new cycle is done.

The so-called Optimized Levels (OL) method was used to determine the wavefunction and energy for each state involved. This allow for a full relaxation of the initial and final states and provide much better energies and wave- functions. However, in this method, spin-orbitals in the initial and final states are not orthogonal, since they have been optimized separately. The formalism to take in ac- count the wavefunctions non-orthogonality in the transi- tion probabilities calculation has been described by L¨ owdin [31] and Slater [32]. The matrix element of a one-electron operator O between two determinants belonging to the initial and final states can be written as

hν Π JM |

N

X

i=1

O (r i ) |ν ^′ Π ^′ J ^′ M ^′ i =

× 1 N !

ψ 1 (r 1 ) · · · ψ m (r 1 ) .. . . .. .. . ψ 1 (r m ) · · · ψ m (r m )

×

m

X

i=1

O (r i )

φ 1 (r 1 ) · · · φ m (r 1 ) .. . . .. .. . φ 1 (r m ) · · · φ m (r m )

, (8)

where the ψ i belong to the initial state and the φ i and primes belong to the final state. If ψ = |nκµi and φ =

|n ^′ κ ^′ µ ^′ i are orthogonal, i.e., hnκµ|n ^′ κ ^′ µ ^′ i = δ n,n

δ κ,κ

δ µ,µ

, the matrix element (8) reduces to one term hψ i | O |φ i i where i represents the only electron that does not have the same spin-orbital in the initial and final determinants.

Since O is a one-electron operator, only one spin-orbital can change, otherwise the matrix element is zero. In con- trast, when the orthogonality between initial and final states is not enforced, one gets [31,32]

hν Π JM |

N

X

i=1

O (r i ) |ν ^′ Π ^′ J ^′ M ^′ i = X

i,j

hψ i | O |φ j

i ξ _ij

D ij

, (9) where D ij

is the minor determinant obtained by crossing out the ith row and j ^′ th column from the determinant of dimension N × N, made of all possible overlaps hψ k |φ l

i and ξ _ij

= ±1 the associated phase factor.

The mdfgme code take into account non-orthogonality for all one-particle off-diagonal operators (hyperfine ma- trix elements, transition rates. . . ). The overlap matrix is

build and stored, and minor determinants are constructed, and calculated using standard LU decomposition.

2.2 Evaluation of QED corrections

In superheavy elements, the influence of radiative correc- tions must be carefully studied. Obviously the status of the inner orbital and of the outer ones is very different. It is not possible for the time being, to do a full QED treat- ment. Here we use the one-electron self-energy obtained using the method developed by Mohr [33,34]. These cal- culations have been extended first to the n = 2 shell [35, 36] and then to the n = 3, n = 4 and n = 5 shells, for

|κ| ≤ 2 [37]. More recently, a new coordinate-space renor- malization method has been developed by Indelicato and Mohr, that has allowed substantial gains in accuracy and ease of extension [38,39]. Of particular interest for the present work, is the extension of these calculation to arbi- trary κ values and large principal quantum numbers [40].

All known values to date have been implemented in the 2006 version of the mdfgme code, including less accurate, inner shell ones, that covers the superheavy elements [41, 42]. The self-energy of the 1s, 2s and 2p 1/2 states is cor- rected for finite nuclear size [43]. The self-energy screening is taken into account here by the Welton method [44,45], which reproduces very well other methods based on direct QED evaluation of the one-electron self-energy diagram with screened potentials [46,47,48]. Both methods how- ever leave out reducible and vertex contributions. These two contributions, however, cancels out in the direct eval- uation of the complete set of one-loop screened self-energy diagram with one photon exchange [49]. The advantages of screening method, on the other hand is that they go be- yond one photon exchange, which may be important for the outer shells of neutral atoms. Recently special studies of outer-shell screening have been performed for alkali-like elements, using the multiple commutators method [50,51].

The comparison between the Welton model and the results from Ref. [50,51] is presented in Table 1. This table confirms comparison with earlier work at lower Z.

It shows that the use of a simple scaling law, as incorpo- rated in GRASP 92 and earlier version of mdfgme does not provide correct values. This scaling law is obtained by comparing the mean value of the radial coordinate over Dirac-Fock radial wave-function hri DF to the hydrogenic one hr (Z eff )i hydr. . This allow to derive an effective atomic number Z eff by solving hr (Z eff )i hydr. = hri DF . One then use Z eff to evaluate the self-energy screening from one- electron self-energy calculations. The superiority of the Welton model can be easily explained by noticing that the range of QED corrections is the electron Compton wave- length Λ C = α a.u., while mean atomic orbital radii are dominated by contributions from the ⁿ _Z

a.u. range, which is much larger.

When dealing with very heavy elements in the limit

Zα → 1, one should consider if perturbative QED is still

a valid model. Unfortunately, we still lack the tools to

answer this fundamental question. However, the use of

Table 1. Self-energy and self-energy screening for element 111

Level 1s 7s

SE (point nucl.) 848.23 3.627 Welton screening − 18.80 − 3.283 Finite size − 50.37 − 0.260 Total SE (DF) 779.06 0.084 Pyykk¨ o et al. [51]

0.087 Pyykk¨ o et al. [51]

0.095 h r i (GRASP) [52]

0.018

a

screening calculated using Dirac-Fock potential

b

screening calculated using Dirac-Slater potential fitted to E

c

use hydrogenic values with Z

obtained by solving h r (Z

) i

= h r (Z

) i

numerical all-order methods may gives some partial an- swers. In particular they allow to include the leading con- tribution to vacuum polarization to all orders by adding the Uehling potential [53] to the MCDF differential equa- tions. This possibility has been implemented in the md- fgme code as described in [54], with the help of Ref. [55].

It also allows to calculate the effect of vacuum polarization in quantities other than energies, like hyperfine structure shifts [54], Land´e g-factors [56], or transition rates. It can also provides some hints on the oder of magnitude of QED effects on atomic wavefunction, orbital or atom radii, and electronic densities.

For high-Z , higher order QED corrections are also im- portant. In the last decade, calculations have provided the complete set of values for two-loops, one-electron diagrams to all orders in Zα [57,58,59,60,61] and also low-Z ex- pansions. All available data has been implemented in the mdfgme code. However, this data is limited to the n ≤ 2 shells.

3 Limitation of the MCDF method

3.1 Non-relativistic limit

The success of relativistic calculations in high-Z elements atomic structure is impressive. It has been shown many times, that only a fully relativistic formalism and the use of a fully relativistic electron-electron interaction can re- produce the correct level ordering and energy in heavy systems. As a non-exhaustive list of example, we can cite the case of the 1s2p ³ P 0 – ³ P 1 level crossing for Z = 47 in heliumlike systems [62,63,64], and the prediction of the 1s ² 2s2p ³ P 0 – ³ P 1 inversion in Be-like iron [65], both due to relativistic effects and Breit interaction. Relativistic ef- fects determine also the structure of the ground configu- ration of many systems, as was recognized for example in the study of lawrencium which has a 7p ² P in place of a 6d ² D configuration[66,67,68].

There is one caveat that must be taken into considera- tion when performing such calculations, that has been rec- ognized in low-Z systems, but never explored in the super-

heavy elements region: it may sound rather paradoxical to investigate the nonrelativistic limit of MCDF and, more generally, of all-order calculations, when studying super- heavy elements. Here we show, however, that there is a problem that has to be taken into account if one wants to obtain the correct fine structure splitting in all cases. We believe it is the first time this problem is recognized in the highly-relativistic limit.

This problem was first found many years ago, in sys- tems like the fluorine isoelectronic sequence [69]: the non- relativistic limit, obtained by doing c → ∞ in a MCDF code, is not properly recovered. States with LSJ label

2S+1 L J and identical L and S, and different J , which should have had the same energy in the non-relativistic limit do not. The energy difference between the levels of identical LS labels but different J is called the non- relativistic (NR) offset. This offset leads to slightly incor- rect fine structure in cases when a relativistic configura- tion has several non-relativistic parents (i.e., several states with one electron less, and different angular structure, that can recouple to give the same configuration). This effect can be large enough to affect comparison between theory and experiment. It should be noted that such a problem does not show if one works in the Extended Average Level (EAL) version of the MCDF. In this case a single wave- function is used for all the members of a given multiplet, and the relaxation effects that are the source of the NR offset disappear, at the price of less accurate transition rates and energies, for a given configuration space.

Very recently, this non-relativistic problem was shown

to be general to any all-order methods, in which sub-

classes of many-body diagrams are re-summed, without

including all the diagrams relevant of a given order. In

the MCDF method, in particular, one should add all con-

figurations with single excitations of the kind nκ → n ^′ κ,

that in principle, should have no effect on the energy in

a non-relativistic calculation, due to Brillouin’s theorem

[70]. In the iso-electronic sequence investigated up to now,

this effects became less severe when going to higher Z, and

it has thus never been considered in very heavy systems: in

neutral, or quasi-neutral superheavy elements, the outer-

shell structure can be complex, with several open shells,

and thus many possible parents core configurations. It is

then worthwhile to study if this problem could arise. We

have studied a number of cases. As a first example we have

studied neutral uranium. The ground state configuration

is known to be [Rn]5f ³ 6d7s ^{2 5} L 6 . We have calculated all

levels of the ground configuration with J = 0 to J = 9,

both in normal conditions, and taking the speed of light

to infinity in the code. The results are shown on Table 2

and Fig. 1. There are two group of levels that can be af-

fected by a NR offset: the 5f ³ 6d7s ^{2 5} H J , J = 3, 4 and the

5f ³ 6d7s ^{2 5} L J , J = 6 to J = 9. The non-relativistic off-

set is evaluated for both groups of levels as the difference

between the energy of a configuration with a given J and

the one for the configuration with the lower energy in the

NR limit. The figure clearly shows a NR offset of 25 meV

for the ⁵ H 4 and ⁵ H 5 , and one of less than 1 meV for the

four ⁵ L J levels. While it can be significant compared to

the accuracy of a laser measurement, it is probably neg- ligible compared to the accuracy of realistic correlation calculations.

In order to assess the generality of this problem, we have investigated several other characteristic systems. El- ement 125, for example, is the first element with a popu- lated 5g orbital [3,4]. We have calculated the NR offset for a configuration with 125 electrons [Rn]5f ¹⁴ 6d ¹⁰ 7s ² 7p ⁶ 8s ² 5g6f ⁴ and Z = 125, 135 and 140. The results are presented on Fig. 2. We find three groups of levels with LSJ labels, four levels with label ⁶ F J (J = 1/2 to J = 7/2), three levels with label ⁶ K J (J = 9/2 to J = 13/2) and four levels with label ⁶ N J (J=15/2 to J = 21/2). The NR offset in each group is of the order of a few meV, while the (non- relativistic) energy difference between the two first groups is 0.19 eV, and between the first and the last groups is around 0.25 eV. The figure also shows that the NR offset gets smaller when Z increases as expected.

Table 2. Total relativistic energy [Ener. (BSC)], including all- order Breit and QED corrections, total non-relativistic energy [ Ener. (NR)], and NR Offset for the ground configuration of uranium, relative to the 5f

6d7s

L

energy, which is the lowest level of the configuration. Correlation has not been in- cluded. One can observe a splitting in the NR energy between the

H levels and the

L levels, which should be exactly de- generate. The fine-structure can be improved by subtract the NR offset from the relativistic energy.

Label Ener. (BSC) Ener. (NR) NR Offset

3

P

1.25373 1.22751

5

D

1.62153 1.22803

5

G

1.28675 0.89836

5

H

0.91680 0.70185 0.0251

5

H

1.05358 0.67678 0.0000

5

K

0.14628 0.01169

5

L

0.00000 0.00000 0.0000

5

L

0.42167 0.00181 0.0018

5

L

0.85283 0.00333 0.0033

5

L

1.28393 0.00284 0.0028

We have also investigated the lower excited states of a somewhat simpler system, element 118 (eka-radon). We have explored the [Rn]5f ¹⁴ 6d ¹⁰ 7s ² 7p ⁵ 8s which should ex- hibit no NR offset (it has single parent states) and [Rn]- 5f ¹⁴ 6d ¹⁰ 7s ² 7p ⁵ 7d. The results are presented on Table 3.

As expected, the 7p ⁵ 8s ³ P J states do not exhibit an NR offset, within our numerical accuracy. The 7p ⁵ 7d ³ L J con- figurations however do have a strong NR offset, up to 0.8 eV, much larger than we expected from the other results presented above, and the largest ever observed.

Clearly, such a large offset would render any calculation of the fine structure splitting of eka-radon useless, unless the results are corrected for the NR offset.

We would like to note that subtracting the NR offset is only a partial fix, since it was shown in Ref. [70] that not only the fine structure is affected, but also the level energy. The only possible solution are thus to do calcu-

[3P0] [5D1] [5G2] [5H3] [5H4] [5K5] [5L6] [5L7] [5L8] [5L9]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Energy (eV)

LSJ Label

BSC offset subtracted NR offset subtracted

Fig. 1. Non-relativistic offset on the ground configuration of uranium. “Ener. (BSC off. sub)”: total MCDF energy, with all QED corrections, using the full Breit operator in the SCF process, relative to the 5f

6d7s

L

energy to which the non- relativistic offset has been subtracted. “Ener. (NR off. sub)”:

Total non-relativistic energy, to which the non-relativistic off- set for members of the same LS multiplet has been subtracted, relative to the 5f

6d7s

L

energy. The two

H

and the four

L

levels have thus identical non-relativistic energy as it should be. Uncorrected energy and NR offset are displayed in Table 2

[6F1/2] [6F3/2]

[6F5/2] [6F7/2]

[6K9/2]

[6K11/2] [6K13/2]

[6N15/2] [6N17/2]

[6N19/2] [6N21/2] 0.000

0.001 0.002 0.003 0.004 0.005

NR offset (eV)

LSJ Label Z=125

Z=130 Z=140

Fig. 2. Non-relativistic offset for the

F

,

K

and

N

LS configurations for the ground configuration of an atom with 125 electron. Z = 125, 130 and 1340 have been evaluated. This contribution should be negligible compared to correlation. For each LS group, the offset is evaluated by subtracting the lower energy of the group to the others.

lations with a large number of configurations, including

all single excitations, or to use the EAL method. In view

of the complexity of calculations on the superheavy ele-

ments, and of the extra convergence difficulties associated

with the presence of Brillouin excitations, the first solution

is probably not very universal, but the second one should

always work. In conclusion, we have shown for the first

time, that it is important to check for the non-relativistic

Table 3. Non-relativistic offset on the lower excited config- urations of eka-radon (Z=118). For each member of a multi- plet, as, e.g., 7p

7d

D, we evaluate the difference between the member with the lowest non-relativistic energy and the others.

Non-relativistically, all member of a multiplet with identical LS labels should have an energy independent of the total angular momentum J.

J 7p

7d

D J 7p

7d

F J 7p

7d

P J 7p

8s

P

1 0.801 2 0.335 0 0.000 0 0.000

2 0.000 3 0.000 1 0.002 1 0.002

3 0.336 4 0.000 2 0.802 2 0.002

limit in super-heavy elements and to correct for the non- relativistic offset, as it can be large in some cases.

3.2 Effect of the all-order Breit operator on simple systems

The use of different form of the electron-electron inter- action in the self-consistent field process has a profound qualitative influence on the behavior of variational cal- culations that goes beyond changes in energy. In partic- ular the mixing coefficients between configurations con- tributing to intermediate coupling are strongly affected (and thus the values of many operators would be like- wise affected). For example, let us examine the very sim- ple case of the 1s2p ³ P 1 state in two-electron system. In a MCDF calculation, intermediate coupling is taken care of by calculating | 1s2p ³ P 1 i = c 1 | 1s2p 1/2 J = 1i + c 2 | 1s2p 3/2 J = 1i. The evolution as a function of the atomic number Z of the c 2 coefficient, is plotted on Fig. 3, with only the Coulomb interaction, or the full Breit interaction made self-consistent. The figure shows clearly that the in- clusion of the Breit interaction in the SCF process lead to values of c 2 that are one order of magnitude lower at high- Z that when only the Coulomb interaction is included. It means that the JJ coupling limit is reached much faster.

This has some influence even in the convergence of the cal- culation: as the exchange potential for the 2p 3/2 orbital is proportional to

c

c

2

, it becomes very large, and the cal- culation does not converge. This can be traced back to a negative energy continuum problem. If we use the method described in Ref. [26] to solve for the 2p 3/2 , then conver- gence can be reached, provided the projection operator that suppress coupling between positive and negative en- ergy solution of the Dirac equation is used. On Fig. 4, the different contributions to the 1s2p ³ P 1 level energy are plotted at Z = 40. The minimum, which corresponds to the level energy, is obtained for a mixing coefficient (ha- bitually obtained by diagonalization of the Hamiltonian matrix), c 2 = 0.104. The shape of the magnetic and re- tardation energy contribution, as observed on the figure, shows clearly that the curve used to find the minimum (which represents the sum of the contribution of the mean values of the operators in Eqs. (3a), Eqs. (3b), and (4)) is shifted to the left compared to the pure Coulomb con- tribution. This explain why the mixing coefficients gets

much smaller when using the Breit interaction in place of the Coulomb interaction in the SCF process. At high-Z this lead to an extra difficulty to achieve convergence: as can be seen in Figs. 5 and 6, the minimum corresponding to the 1s2p ³ P 1 state (the one lower in energy) is very close to c 2 = 0. It thus sometimes happens during the convergence, that c 2 changes sign, leading to very tedious tuning of the convergence process. This is even worse for the 1s2p ¹ P 1 state, because one is trying to reach the maximum energy. In that case the oscillation of the c 1 co- efficient around zero are impossible to damp. Obviously, such problems will slowly disappear when going to neutral systems. For example, in neutral nobelium (Z = 102) c 2

changes only from 0.47250012 to 0.47316712.

10 20 30 40 50 60 70 80 90 100 110

10^-3 10^-2 10^-1 10⁰

Mixing coeffcient

Z

BSC BP

Fig. 3. Variation with Z of mixing c

coefficient for the 1s2p

P

level of helium-like ions. BP: Only the Coulomb in- teraction is used in the SCF process. BSC: The full Breit in- teraction is used in the SCF process

4 Relativistic and QED effects on transition energies and probabilities

4.1 Beryllium isoectronic sequence correlation

It is interesting to investigate simple many-body system,

that can be calculated accurately, to see which kind of

highly relativistic effects can be expected in the limit Zα →

1. In that sense the beryllium isoelectronic sequence is an

interesting model case, as it exhibit a very strong intrashell

coupling between the 1s ² 2s ² J = 0 and 1s ² 2p ² _1/2 J = 0 con-

figurations, which are almost degenerate in energy. We

thus calculated all contributions to the energy, with and

without including the vacuum polarization and Breit in-

teraction in the SCF process. From this we could deduce

the loop-after-loop Uehling contribution to the total en-

ergy, and the intrashell correlation. Most quantities con-

tributing to the total energy do not exhibit any specific

behavior when Zα → 1. However, a major changes in be-

havior of the system occurs around Z = 125 as shown

-1.0 -0.5 0.0 0.5 1.0 -2

-1 0 1 2 3 4 5

-2.760x10

-2.758x10

-2.756x10

-2.754x10

-2.752x10

-2.750x10

-2.748x10

-2.746x10

-2.744x10

-2.742x10

Ma g n e ti c a n d R e ta rd a ti o n e n e rg y (e V)

c 2

Mag.

Ret.

H. O. Ret.

Coul Total e-e Total

C o u lo mb a n d To ta l e n e rg y (e V)

Fig. 4. Variation of the different contributions to the energy as a function of the mixing coefficient c

, for the 1s2p

P

level at Z = 40. The arrow indicates the position of the 1s2p

P

energy, at the minimum around c

= 0.104. Left axis: “Mag.”:

Magnetic energy, Eq. (3b). “Ret.”: Breit retardation, Eq. (4). “H.O. Ret.”: higher-order retardation. Right axis: “Coul.”:

Coulomb energy Eq. (3a). “Total e-e”: sum of the 4 preceding contributions. “ Total”: total level energy including all QED corrections.

in Fig. 7. One can see that the ground state, which is 1s ² 2s ² J = 0 at lower Z, becomes 1s ² 2p ² _1/2 J = 0. This can be seen on the mixing coefficients as plotted on Fig.

7. This translates into a strong increase in the loop-after- loop vacuum polarization contribution. Obviously, if we were able to evaluate other second-order QED calcula- tion than loop-after-loop vacuum polarization, including off-diagonal two-electron self-energy matrix elements for quasi-degenerate state, following recent work on helium- like systems [71,72,73], there could be more unexpected effects to observe.

While not displaying such a feature, the total corre- lation energy increases strongly, reaching up to 3.6 keV.

One can observe effects on other properties of the atom, like orbital energies and mean orbital radius. Figure 8 shows that, in the same atomic number range when the ground state changes of structure, the 2p 3/2 orbital radius and energy exhibit a very strong change. The behavior of the ground state must be connected to the fact that the small component of 2p 1/2 orbitals, as can be seen from

Eq. (7), has a s behavior, and the ratio between small and large component is of order Zα. It is thus under- standable that such effects could occurs when Zα → 1. It should be noted that this effects happens even with a pure Coulomb electron-electron interaction, which is one more proof it is only connected with the behavior of the one- electron wavefunctions. We investigated the similar case of the magnesiumlike sequence, which exhibit strong in- trashell coupling between the 3s, 3p and 3d orbitals, but we could not observe any effect on energies in this range of Z. However the [Ne]3p ² _1/2 mixing coefficient started to increase faster around Z = 128, but convergence problems prohibited us to investigate higher Z.

4.2 Relativistic correlations on the neon isoelectronic sequence

Calculating completely correlated energies can be performed

only on relatively small systems. Neonlike ions, with 10

-1.0 -0.5 0.0 0.5 1.0 -40

-30 -20 -10 0 10 20 30 40 50 60

-1.66x10

-1.65x10

-1.64x10

-1.63x10

-1.62x10

-1.61x10

-1.60x10

Ma g n e ti c a n d R e ta rd a ti o n e n e rg y (e V)

c 2

Mag.

Ret.

H. O. Ret.

Coul Total e-e Total

C o u lo mb a n d To ta l e n e rg y (e V)

Fig. 5. Variation of the different contributions to the energy as a function of the mixing coefficient c

, for the 1s2p

P

level at Z = 92. The arrow indicates the position of the 1s2p

P

energy, at the minimum around c

= 0.002. Left axis: “Mag.”:

Magnetic energy, Eq. (3b). “Ret.”: Breit retardation, Eq. (4). “H.O. Ret.”: higher-order retardation. Right axis: “Coul.”:

Coulomb energy Eq. (3a). “Total e-e”: sum of the 4 preceding contributions. “ Total”: total level energy including all QED corrections.

electrons are a small enough system that can be calcu- lated with rather large basis sets. We have extended the calculation performed in Ref. [7] to superheavy elements (Z = 134). All 10 electrons are excited to all virtual or- bitals up to a maximum nℓ, that is varied from 3d to 6h.

The results are presented in Table 4 and plotted on Fig. 9.

The trend found up to Z = 94 in Ref. [7] extend smoothly to larger Z, but is enhanced. The total correlation energy becomes very large. It doubles when going from Z = 95 to Z = 134, the highest Z for which convergence could be reached. The speed of convergence as a function of nℓ does not change with increasing Z.

4.3 Transition energies and probabilities in nobelium and element 118 (eka-radon)

In this section we study the different contributions to the energy and transition probabilities of nobelium (Z = 102) and eka-radon (Z = 118). Nobelium is the next candidate for measurement of its first excited levels transition en-

Table 4. Total correlation energy for neonlike ions, with the Breit interaction included in the SCF, as a function of the most excited orbital included in the basis set.

Z all → 3d all → 4f all → 5g all → 6h

10 − 5.911 − 8.306 − 9.339 − 9.709

15 − 5.989 − 8.712 − 9.838 − 10.280

25 − 6.374 − 9.310 − 10.494 − 10.967

35 − 6.710 − 9.850 − 11.099 − 11.609

45 − 7.074 − 10.482 − 11.816 − 12.372

55 − 7.515 − 11.269 − 12.710 − 13.322

65 − 8.067 − 12.260 − 13.833 − 14.511

75 − 8.752 − 13.375 − 15.247 − 16.007

85 − 9.772 − 15.119 − 17.056 − 17.916

95 − 11.160 − 17.231 − 19.429 − 20.415

105 − 13.205 − 20.146 − 22.700 − 23.853

114 − 15.975 − 23.966 − 26.974 − 28.332

124 − 21.222 − 30.897 − 34.694 − 36.389

130 − 26.714 − 37.777 − 42.304 − 44.300

134 − 32.248 − 44.412 − 49.598 − 51.860

-1.0 -0.5 0.0 0.5 1.0 -40

-30 -20 -10 0 10 20 30 40 50 60

-1.66x10

-1.65x10

-1.64x10

-1.63x10

-1.62x10

-1.61x10

-1.60x10

Ma g n e ti c a n d R e ta rd a ti o n e n e rg y (e V)

c 2 Mag.

Ret.

H. O. Ret.

Coul Total e-e Total

C o u lo mb a n d To ta l e n e rg y (e V)

Fig. 6. Variation of the different contributions to the energy as a function of the mixing coefficient c

, for the 1s2p

P

level at Z = 92. The arrow indicates the position of the 1s2p

P

energy, at the maximum around c

= 0.002. Left axis: “Mag.”:

Magnetic energy, Eq. (3b). “Ret.”: Breit retardation, Eq. (4). “H.O. Ret.”: higher-order retardation. Right axis: “Coul.”:

Coulomb energy Eq. (3a). “Total e-e”: sum of the 4 preceding contributions. “ Total”: total level energy including all QED corrections.

ergy by laser spectroscopy. Large scale calculations have been performed recently [21]. Here we examine a number of corrections not considered in Ref. [21]. For the 7s ^{2 1} S 0

ground state, we did a MCDF calculation taking into ac- count all single and double excitations, except the one corresponding to the Brillouin theorem, from the 5f and 7s shells to the 7p, and 6d shells (48 jj configurations). For the excited states we included excitations from 5f , 7s and 7p shells to the 7p and 6d ones. This leads to 671 jj con- figurations for the 7s7p ³ P 1 and 7s7p ¹ P 1 states and 981 for 7s7p ³ P 2 . The results of this calculation are presented on Table 5 for the transition energies and on Table 6 for transition probabilities. It is clear from these two tables that many contributions that are important for level en- ergies are completely negligible for transition probabilities in this case. This is not true for highly charged ions, even at much lower Z . In particular two-loop QED corrections, even though many of them have not been calculated from n > 2, should remain completely negligible, since they will be of the same order of magnitude as the loop-after-

loop and K¨ all`en and Sabry contributions. The self-energy screening is almost exactly compensating the self-energy, but leaves a contribution that should be visible if one can calculate correlation well enough. At the present level of accuracy, calculation of the pure Coulomb correlation is the real challenge as it requires considerable effort on the size of configuration space.

The transition probabilities on Table 6 have been eval- uated with different approximation, using theoretical en- ergies. Here the choice of the wavefunction optimization technique has a sizeable effect. It remains small compared to correlation, but still at the level of 0.2%. We also inves- tigated the effect of using fully relaxed orbitals on both initial and final state. In particular we checked at the Dirac-Fock level of approximation, what is the order of magnitude of taking into account non-orthogonality be- tween initial and final state orbitals. We found 2.2% for the ³ P 1 → ¹ S 0 transition, 0.014% for the ¹ P 1 → ¹ S 0

transition and -0.025% for the ³ P 2 → ¹ S 0 . For calcula-

tions between correlated wavefunctions, it is anyway im-

80 90 100 110 120 130 10^-1

10⁰ 10¹ 10² 10³ 10⁴

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

E (eV)

Z

L-a-L Uelhing Correlation

c2 i

c 1

2

c 3

2

c3 2

Fig. 7. Loop-after-loop Uehling contribution to berylliumlike ions ground state energy (changed of sign), obtained by includ- ing the Uehling potential in the SCF. The intrashell correlation energy is also plotted (left axis), as well as the square of the mixing coefficients of the 1s

2s

J = 0, 1s

2p

J = 0 and 1s

2p

J = 0 configurations (right axis).

80 90 100 110 120 130 140

0.0 5.0x10⁴ 1.0x10⁵ 1.5x10⁵ 2.0x10⁵ 2.5x10⁵ 3.0x10⁵ 3.5x10⁵ 4.0x10⁵

0.00 0.01 0.02 0.03 0.04 0.05

Orbital energy (eV)

c2 OE(1s) OE(2s) OE(2p_1/2) OE(2p_3/2)

r(1s) r(2s) r(2p_1/2) r(2p_3/2)

Orbital radii (a.u.)

Fig. 8. Orbital radii (right axis) and one-electron energies (left axis) for Be-like ions

portant to evaluate the matrix elements for off-diagonal operators, taking account non-orthogonalities between ini- tial and final state orbitals, as it can change dramatically the contribution of a given CSF, since overlaps between correlations orbitals in initial and final states can be very different from either one or zero.

As another example we have also calculated several transition energies and rates for element 118, which are displayed in Table 7. For the most intense 7p ⁵ 7d → 7s ² 7p ⁶ transitions and for the 7p ⁵ 8s → 7s ² 7p ⁶ transitions we did a MCDF calculation, taking into account all single and double excitations, (except the ones corresponding to the Brillouin theorem) from the 7s and 7p shells to the 7d and 6f shells for the 7p ^{6 1} S 0 ground state (38 jj configurations) and for the 7p ⁵ 7d excited states (657 jj configurations),

10 20 30 40 50 60 70 80 90 100 110 120 130 140 -55

-50 -45 -40 -35 -30 -25 -20 -15 -10 -5

Correlation energy (eV)

Z all -> 3d all -> 4f all -> 5g all -> 6h

Fig. 9. Total correlation energy for neonlike ions, with the Breit interaction included in the SCF, as a function of the most excited orbital included in the basis set.

and from the 7p and 8s shells to the 7d and 6f shells for the 7p ⁵ 8s excited states (151 jj configurations).

The transition energy was calculated for some of these transitions, with and without including the vacuum polar- ization and Breit interaction in the SCF process, and we concluded that the transition energy is not significantly affected by the inclusion of these interactions in the SCF.

This is illustrated on Fig. 10 that shows the transition energy values for 7p ⁵ 7d ³ P 1 transition.

6.81450 6.81455 6.81460 6.81465 6.81470 6.81475 6.81480 6.81485 6.81490

Transition energy (eV)

BSC, VPSC BSC, VPP BP, VPP

Fig. 10. Changes in the 7p

8s

P

→ 7s

7p

S

transition

energy with respect to the approximation made, for element

118. BSC: Breit self-consistent. VPSC: Uelhing potential self

consistent. BP: Breit in perturbation. VPP: Uelhing vacuum

polarization in perturbation.

Table 5. Transition energies for the lower energy levels of nobelium (eV). Coul.: DF Coulomb energy. Mag. (pert), Ret.

(pert.), Higher order ret. (pert.): Contribution of the Magnetic, Breit retardation and higher-order retardation in first order of perturbation. All order Breit: effect of including the full Breit interaction in the SCF. Coul. Corr.: Coulomb correlation.

Breit Corr.:Contribution of all Breit terms to the correlation energy. Self-energy (FS): self-energy and finite size correction.

Self-energy screening: Welton approximation to self-energy screening. Vac. Pol. (Uehling): mean value of the Uehling potential (order α(Zα)). VP (muons, Uehling): vacuum polarization due to muon loops. VP Wichman and Kroll: correction to the Uehling potential (order α(Zα)

). Loop-after-loop V11: iterated Uehling contribution to all orders. VP (K¨ all`en et Sabry): two- loop contributions to vacuum polarization. Other 2nd order QED: sum of two-loop QED corrections not accounted for in the two previous one. Recoil: sum of lowest order recoil corrections (see, e.g., [74]). The number of digits presented in the table is not physically significant, but is necessary to show the size of some contributions. The physical accuracy is not better than 1 digits.

Contribution

P

→

S

P

→

S

P

→

S

Coul. 1.69909 3.20051 2.19931

Mag. (pert) 0.00070 − 0.00420 − 0.00347

Ret. (pert.) − 0.00048 0.00005 − 0.00058

Higher order ret. (pert.) − 0.00119 0.05048 − 0.00231 All order Breit (pert) 0.00002 − 0.05877 0.00003

Coul. Corr. 0.59560 0.60414 0.55545

Breit Corr. − 0.00208 − 0.00229 − 0.00194

Self-energy (FS) − 1.60884 − 1.72025 − 1.76085 Self-energy screening 1.59030 1.72940 1.74228 Vac. Pol. (Uehling) 0.00610 0.00435 0.00648 VP (muons. Uehling) 0.00000 0.00000 0.00000 VP Wichman and Kroll − 0.00030 − 0.00019 − 0.00033 Loop-after-loop Uehl. 0.00002 0.00012 0.00003 VP (K¨ all`en and Sabry) 0.00004 0.00002 0.00005 Other 2nd order QED 0.00000 0.00000 0.00000

Recoil − 0.00082 − 0.00081 − 0.00082

Total 2.28 3.80 2.73

Ref. [21] II 2.34 3.49

Ref. [21] I 2.60 3.36

Table 6. Effect of correlation, Breit interaction and all order vacuum polarization on transition probabilities of nobelium. DF:

Dirac-Fock. VP: vacuum polarization. ex.: excitation

initial 7s7p

P

DF 7s7p

P

ex. 7s, 7p → 7p, 6d 7s7p

P

ex. 5f, 7s, 7p → 7s, 7p, 6d final → 7s

S

DF → 7s

S

ex. 7s → 7p, 6d → 7s

S

ex. 7s → 7p, 6d 7s7p

P

→ 7s

S

(E1)

Breit Pert.; VP. Pert. 1.071 × 10

3.520 × 10

1.947 × 10

Breit Pert.; VP SC 1.073 × 10

3.526 × 10

1.954 × 10

Breit SC; VP Pert. 1.062 × 10

3.501 × 10

1.920 × 10

Breit SC; VP SC 1.064 × 10

3.507 × 10

1.927 × 10

7s7p

P

→ 7s

S

(E1)

Breit Pert.; VP. Pert. 2.705 × 10

3.500 × 10

3.680 × 10

Breit Pert.; VP SC 2.697 × 10

3.559 × 10

3.738 × 10

Breit SC; VP Pert. 2.711 × 10

3.509 × 10

3.686 × 10

Breit SC; VP SC 2.704 × 10

3.568 × 10

3.743 × 10

7s7p

P

→ 7s

S

(M2)

Breit Pert.; VP. Pert. 1.717 × 10

5.187 × 10

5.492 × 10

Breit Pert.; VP SC 1.713 × 10

5.173 × 10

5.477 × 10

Breit SC; VP Pert. 1.721 × 10

5.200 × 10

5.513 × 10

Breit SC; VP SC 1.717 × 10

5.187 × 10

5.499 × 10

Table 7. Transition energies and transition rates to the ground state 5f

6d

7s

7p

S

of eka-radon (element 118). The transition values from states labeled with

were calculated with correlation up to 6f, as well as the ground state.

Initial State Transition Energy Transition Rate

(eV) (s

)

5f

6d

7s

7p

7d

D

+

P

10.43 9.86 × 10

3

P

6.81 5.53 × 10

3

D

+

P

7.2 9.98 × 10

3

P

6.08 1.41 × 10

3

F

16.95 1.11 × 10

3

F

6.15 1.30 × 10

1

D

6.25 1.10 × 10

3

D

6.22 7.52 × 10

3

F

6.01 3.91 × 10

3

P

6.64

5f

6d

7s

7p

8s

P

4.73 2.04 × 10

3

P

4.3 2.04 × 10

5 Relativistic and QED effects on Land´ e factors, atomic radii and electronic densities

Although transition energies and probabilities, as well as ionization potential, are important quantities, it is in- teresting to study relativistic and QED effects on other atomic parameters, like Land´e g-factors, atomic radii and electronic densities. Land´e factors, which define the strength of the coupling of an atom to a magnetic field, can help characterize a level.

In some experiments concerning superheavy elements, singly ionized atoms are drifted in a gas cell, under the in- fluence of an electric field. The drift speed can be related, in first approximation, to the charge distribution radius of the ion [75,76]. In this context, it can also be interesting to look at the atomic density, and see how it is affected by relativistic and QED effects. In that case, though, we can only get a feeling of this effect by comparing den- sities calculated with and without the Breit interaction self-consistent, or with and without the Uehling potential included in the differential equation. There is currently no formalism that would enable to account for changes in the wavefunction related to the self-energy.

We define the radial electronic density as r ² ρ (r) =

Z

dΩr ² ρ (r, Ω )

= X

i ∈ occ.orb.

̟ i

P i (r) ² + Q i (r) ²

, (10) where P and Q are defined in Eq. (7), and the ̟ i are the orbital effective occupation numbers,

̟ i = X

j

c ² _j n ⁱ _j , (11) where the sum extend over all configuration containing orbital i, and n ⁱ _j is the occupation number of this or- bital in the configuration j. The density is normalized to R ∞

0 drr ² ρ (r) = N e , where N e is the number of electrons in the atom or ion.

The effect of the Breit interaction and vacuum polar- ization on the charge density of the ground state of Fm ⁺ ([Rn]5f ¹² 6s) is shown on Fig. 11. The inclusion of both contributions leads to local changes of around 1% in the charge density. It is rather unexpected that both contri- bution extend their effects way pass their range, in par- ticular for the vacuum polarization potential, which has a very small contribution for r > 1/α = 0.0073 a.u.

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹ 10² 10³

Ur2 (a.u)

r (a.u.)

Density Diff (Bsc) Diff (Vsc) <r>

msr R_max(7s)

Fig. 11. Total electronic density of Fm

. The vertical lines represents, from left to right the mean radius, the mean spher- ical radius (“rms”) and the mean radius of the outermost or- bital (7s). “Density”: total electronic density Eq. (10). “Diff (Bsc)”: absolute value of the variation in the density due to the inclusion of the Breit interaction in the SCF. “Diff (Vsc)”:

absolute value of the variation in the density due to the in- clusion of the Uehling potential in the differential equations (NB: the dip in the curve correspond to sign changes of the correction).

Charge distribution can be described in a number of

ways, and the definition of atomic or ionic radius has

evolved over the years. Slater noticed a long time ago the

correlation between the value of the maximum charge den- sity of the outermost core electron shell and the ionic ra- dius of an atom [77]. Such a definition, however, does not provide a way to take into account mixing of outer shell in MCDF calculations. Other authors have chosen either the mean radius of a specific orbital hr n,l,j i [78] or weighted mean radius [79]. Results have been obtained for bohrium and hassium. It is not at all obvious either than what is true in crystals must be valid for singly charged ions drift- ing in a gas. Here we have evaluated four different quan- tities and their dependence in the Breit interaction and vacuum polarization. We have evaluated the position of the maximum density of the outermost orbital, the mean radius of the outermost orbital hr n,l,j i, the atom mean ra- dius, and the atom mean spherical radius. The two first quantities have been tested in detail, but they cannot rep- resent the ionic radius when one has a complex outer shell structure or when one calculate correlation. The mean ra- dius of the atom is represented as

hr ^(p) _at. i = 1 N e

Z ∞

0

drr ^p r ² ρ (r)

= 1 N e

X

i∈occ.orb.

̟ i hr ^(p) _n

_,l

_,j

i (12) for p = 1, and the mean spherical radius is obtained as q

hr ⁽²⁾ _at. i. Both can be calculated in a MCDF model, as the

̟ i contain both the occupation numbers and the mixing coefficients. As an example, we have plotted the contri- bution to hr _at. ^(p) i to Sg ⁺ on Fig. 12 of individual values of ̟ i hr ^(p) _n

_,l

_,j

i for the different orbitals. Contrary to the individual orbitals quantities or to the average performed only on the outer shell, like in Ref. [9], both quantities have sizeable contributions from several outer shells. The 7s, 6p, 5f and 6d contribute significantly to both the mean and the mean spherical atomic radius, but the contribu- tion of the 6d orbitals is more dominant for the mean spherical radius. The results for singly charged ions with 57 ≤ Z ≤ 71 and 90 ≤ Z ≤ 108 are presented on Table 8 and plotted on Fig. 13. The comparison between mean ra- dius and mean spherical radius with the Breit interaction in the SCF process with Coulomb values, shows changes around 0.04% to 0.08% depending on the atomic number.

Self-consistent vacuum polarization has an effect 2 to 4 times smaller, depending on the element. We emphasize the fact that we used for the ion the configuration corre- sponding to the ground state of a Dirac-Fock calculation (i.e., without correlation), as given in [6]. For a few ele- ments, the physical ground state configuration, as given for example on the NIST database, is different.

We have used the definition above to evaluate the effect of correlation on the ion radii. The calculation has been performed on neutral nobelium (Z = 102). The results are presented on Table 9. One can see that the radius of max- imum density follows exactly the trend of the mean radius of the outer orbital. However the atomic mean radius and mean spherical radius follow a different trend.

There has been two experiments measuring drift time of singly ionized transuranic elements in gases. One mea-

sured the relative velocity of Am ⁺ with respect to Pu ⁺ [75]

and an other one measured the same quantity for Cf ⁺ and Fm ⁺ [18]. The results of these experiments are expressed as

∆r A − B = r A − r B

r B

. (13)

The experiments above provide ∆r _Am

_{− Pu}

=-3.1 ± 1.3%

and ∆r _Fm

_−Cf

=-2% respectively, which shows a shrink- age of Am as compares to Pu and of Fm as compared to Cf. The comparison between these experiments and the calculation of singly charged ion radii from Table 8 is shown in Table 10. Although it is difficult to draw any firm conclusion from so few data, it seems that

q

hr _at. ⁽²⁾ i re- produces best the experiment, followed by hr _at. ⁽¹⁾ i. We want to point out that the < r 7s > values for neutral atom, from Ref. [5], or ours, which are in good agreement gives a - 1.5% change for Am/Pu and -3.0% for Fm/Cf, compare well with experiment. The 7s radius of the singly charged ions, as shown in Table 8, increases when going from Am to Pu, thus leading to a ratio of the wrong sign, while the 5f average radius, or the global atomic radii as defined here all show the right trend. The change of behavior for Am, is due to the relative diminution of the l(l + 1)/r ² barrier compared to the Coulomb potential as a function of Z . The 5f radius reduces then strongly, leading to the change of the structure of the ground configuration, and to the change of the radius of the 7s orbital.

1s 2s 3s4s 5s 6s 7s

2p* 3p*

4p* 5p*

6p*

2p 3p 4p 5p6p

3d* 4d*

5d* 6d*

3d 4d 5d 6d4f*5f* 4f 5f

Total 0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

<r>×occupation / Ne (a.u.)

Orbital label <r>

<r²>

Fig. 12. Contribution to the mean radius h r

i to Sg+ of individual values of ̟

h r

i,li,ji

i for the different orbitals and p = 1 and 2, together with the total values on the right. Stared orbital labels correspond to the orbital with j = l −

.

QED corrections on Land´e factors of one and three

electron atoms have been studied in great details in the

last few years, because of the increase in experimental

accuracy associated with the use of Penning traps in a

new series of experiments [80,81]. These experiments pro-

vided a strong incentives to evaluate very accurately QED

corrections on Land´e factors, beyond the one due to the

anomalous magnetic moment of the electron [82,83,84,